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## Algebra 2

### Course: Algebra 2>Unit 12

Lesson 4: Modeling with two variables

# Rational equation word problem

Sal models a context that concerns the number and price of pizza slices. The model turns out to be a rational equation. Created by Sal Khan.

## Want to join the conversation?

• at how do I know which set of numbers to add/subtract from each side? Real confused about it :/ and how do you know weather to add/subtract? • In the upcoming practice section, Modeling with one-variable equations and inequalities, I stumbled across a problem which, given the following equation, required us to solve for t:

1024*(1/2)^(t/29)=32
solve for t.
The solution given in the hints went:
(1/2)^(t/29)=32/1024
(1/2)^(t/29)=1/32
(1/2)^(t/29)=(1/2)^5
therefore
t/29=5
t=29*5=145

Now, although I fully understand the logic when I see it, the passage that went from (1/2)^(t/29)=1/32 to (1/2)^(t/29)=(1/2)^5 would never have occurred to me on my own, not in a million years. Is there any previous section I should study harder, any method of handling bases and exponents I should assimilate perhaps, or am I just not smart enough to see what should be obvious at this stage? Thanks in advance for any answer. • If you have a variable or constant in an exponent that you need to solve for, you only have a few options available to you at this level of study (there are some much more advanced techniques, but those a few years ahead of Algebra 2). And, actually, the two methods are really just different applications of the same method.

Method 1: use logarithms to bring the constant or variable out of the exponent, so you can solve for it.
Method 2: get both sides of the equation to be the same base raised to some exponent. If you can do that, then you know that the two exponents must be equal to each other.

You've seen method 2, but it is not always particularly easy to use (and is just a different way of doing method 1 anyway). So, let us look at how to do this same problem with method 1:
1024(½)^(t/29)=32
(½)^(t/29)=32/1024
(½)^(t/29)= 1 /32
log [(½)^(t/29)] = − log (32) ←This uses the property log (1/a) = − log (a)
(t/29) log (½) = − log (32) ← This uses the property log (a^b) = b log (a)
(t/29)[− log 2] = − log (32)
(t/29) = − log (32)] / (− log 2)
t/29 = 5
t = 29*5
t = 145
• Why is it that making more pizzas per day would result in her expenses not changing when the ingredients started costing more? Maybe I'm just overthinking it but I don't really get it. • A slow train traveling from Tashkent to Samarkand arrives 9 minutes late when traveling at 36km/h. If it travels at 27km/h it arrives 39 minutes late. What is the distance between Tashkent and Samarkand?
Help me, please, simulate this problem, if anyone can! • At , for the equation on the right (purple), can't it be simplified to 8+2?

My reasoning: 8+2(p+8)/p+8, the p+8 cancel out and left with 8+2.

Also at , does the coefficient must be 1? • No becasue it is (8+2(p+8)/(p+8), so you are not canceling equivalents. If you broke this fraction down into 2 parts, it would be 8/p+8 + 2. This makes it worst than where we started.
To your second question, a coefficient of 1 can become invisible, so p^2 and 1 p^2 are actually the same thing. If you see a variable without a coefficient, get used to the idea that there is an invisible 1 in front that will almost never show up in final answers.
• At , why does Sal divide the whole "8 + 1.5p"
expression by p and why did we not divide just the "1.5p" by p to get the cost for per pizza? • 1.5p/p only tells us the price of toppings per pizza, which we already know. We want the total price per pizza in one day. So for 1 pizza running the stove is 8 dollars and the price of toppings is 1.5, this means we can estimate that one pizza cost 9.50 dollars. similarly, 2 pizzas would be 8 + 1.5*2 = 11 dollars. this means 11/2 = 5.50 dollars per pizza. It not TECHNICALLY the actual price per pizza, but it is a helpful way to express it. Does this help?
• p=-16 is called extraneous, isn't it ? Also shouldn't we assume P doesn't equal 0 or -8 ? because if it's we would've divided by 0 also would've multiplied by zero which loose information • why do we need to find per pizza?

Especially in this equation

Dominic from "Dominic's Pizza" always bakes p pizzas each day. Currently, it costs him \$10 per day to use a brick oven and \$2 per pizza for the ingredients.
One day Dominic realized that if he switched to an electric oven, the use of the oven would cost him \$20 per day, but the ingredients would be cheaper, only \$0.80 per pizza. This way, his total expenses for each pizza (including shared oven costs and ingredient costs) would be reduced by \$1   